Math 570: Topology I

Colorado State University, Fall 2017

Instructor: Henry Adams
Email: henry dot adams at colostate dot edu
Office: Weber 125
Office Hours: Tuesday 12-1pm and Wednesday 10-11am in Weber 125, or by appointment.

Lectures: MWF 9:00-9:50am in Weber 202
Textbook: Introduction to Topological Manifolds by John Lee (Second Edition).
An electronic copy of this book is freely available to CSU students here.

Overview: We plan to spend the first three weeks of the class on point-set topology (topologies, continuity, quotient spaces, connectedness, compactness). The goal for the rest of the course will be to introduce algebraic topology, while returning to topics from point-set topology as they arise. During the first third of the course, we hope to introduce the very basics of homotopy equivalences, the fundamental group, higher homotopy groups, and category theory. The second third of the course will focus on homology, with simplicial homology before singular homology. In the final third of the course we return to the fundamental group (Seifert-Van Kampen Theorem), covering maps, and other selected topics (some skipped along the way).

Though I hope to cover many of the topics in our textbook, I will cover some of them (such as portions of point-set topology, and the classification of compact surfaces) in a very casual way: I will provide intuition for the main theorems instead of giving complete proofs. Other topics, such as homology, will be emphasized and presented in detail.

Syllabus: Here is a course syllabus.

Notes

Scans of Henry's lecture notes.

Homework

Homework 1 (LaTeX Source) is due Friday, August 25.
Homework 2 (LaTeX source) is due Friday, September 1.
Homework 3 (LaTeX source) is due Friday, September 8.
Homework 4 (LaTeX source) is due Friday, September 15.
Homework 5 (LaTeX source) is due Friday, September 22.
Homework 6 (LaTeX source) is due Friday, October 6.
Homework 7 (LaTeX source) is due Friday, October 13.
Homework 8 (LaTeX source) is due Friday, October 20.
Homework 9 (LaTeX source) is due Friday, October 27.
Homework 10 (LaTeX source) is due Friday, November 10.
Homework 11 (LaTeX source) is due Friday, November 17.
Homework 12 (LaTeX source) is due Friday, December 1.

We will have weekly homework assignments. All homework is due in class at the beginning of class. Your homework should be legible and stapled.

Exams

The exams will be in-class. You will only be able to use your brain and a pen or pencil - no notes, books, or electronic devices.

Here is Practice Midterm 1.
Here is Practice Midterm 2.
Here is a Practice Final.

Preparation Materials

One way to prepare for this course is by using the materials available at my class webpage Math 472, Fall 2016. Class notes, homework problems, and practice exams are available there. If you email me, I can also send you homework solutions and practice exam solutions. Not all of this material is assumed for Math 570, but we will be covering much of it at a fairly quick pace.

Schedule

Date	Topic	Remark

Aug 21	Course overview & Chp 1
Aug 23	Chp 2: Topologies, Convergence and Continuity
Aug 25	Chp 2: Hausdorff Spaces, Bases and Countability	Homework 1 due

Aug 28	Class cancelled :(
Aug 30	Chp 2: Manifolds
Sep 1	Chp 3: Subspaces and Product Spaces	Homework 2 due

Sep 4	No class
Sep 6	Chp 3: Disjoint Union Spaces, Chp 7: Categorical Products	Last day to drop or change grading option
Sep 8	Chp 7: Categorical Coproducts, Chp 3: Quotient Spaces	Homework 3 due

Sep 11	Chp 4: Connectedness
Sep 13	Chp 4: Compactness
Sep 15	Chp 4: Compactness	Homework 4 due

Sep 18	Chp 7: Homotopy
Sep 20	Chp 7: The Fundamental Group
Sep 22	Chp 7: The Fundamental Group	Homework 5 due

Sep 25	Chp 7: The Fundamental Group
Sep 27	Review
Sep 29	Midterm #1 starting at 8:30am

Oct 2	Chp 7: Homomorphisms Induced by Continuous Maps
Oct 4	Chp 7: Homomorphisms Induced by Continuous Maps
Oct 6	Chp 7: Homomorphisms Induced by Continuous Maps	Homework 6 due

Oct 9	Chp 7: Categories and Functors
Oct 11	Chp 7: Homotopy Equivalence
Oct 13	Chp 7: Higher Homotopy Groups	Homework 7 due

Oct 16	Chp 5: Cell Complexes and CW Complexes	End of course withdrawal period
Oct 18	Chp 5: Simplicial Complexes
Oct 20	Class cancelled; read Theorem 7.40 & Lemma 7.45	Homework 8 due

Oct 23	Chp 13: Simplicial Homology
Oct 25	Chp 13: Simplicial Homology
Oct 27	Chp 13: Simplicial Homology	Homework 9 due

Oct 30	Chp 13: Singular Homology
Nov 1	Review
Nov 3	Midterm #2 starting at 8:00am

Nov 6	Chp 13: Singular Homology
Nov 8	Midterm 2 and Homework 10 comments
Nov 10	Chp 13: Singular Homology	Homework 10 due

Nov 13	Chp 13: Functoriality of simplicial homology
Nov 15	Chp 13: Difference between homotopy groups and homology
Nov 17	Chp 13: The Mayer-Vietoris Theorem	Homework 11 due
Fall Recess, Nov 20-24
Nov 27	Chp 13: Homotopy invariance
Nov 29	Chp 13: Homotopy invariance
Dec 1	Chp 13: Mayer-Vietoris example	Homework 12 due

Dec 4	Chp 13: Degree theory for spheres
Dec 6	Chp 13: Degree theory for spheres
Dec 8	Review


Final Exam, Monday December 11
7:30-9:30am in Weber 202